The Category of $$\omega $$-Effect Algebras: Tensor Product and $$\omega $$-Completion

Abstract Effect algebras are certain ordered structures that serve as a general framework for studying the algebraic semantics of quantum logic. We study effect algebras which obtain suprema of countable monotone sequences – so-called $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -effect algebras. This assumption is necessary to capture basic (non-discrete) probabilistic concepts. We establish a free $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -completion of effect algebras (i.e., a left adjoint to the functor that forgets the existence of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -suprema) and the existence of a tensor product in the category of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -effect algebras. These results are obtained by means of so-called test spaces. Test spaces form a category that contains effect algebras as a reflective subcategory, but provides more space for constructions.